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Theorem efrirr 5570
Description: A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
efrirr ( E Fr 𝐴 → ¬ 𝐴𝐴)

Proof of Theorem efrirr
StepHypRef Expression
1 frirr 5566 . . 3 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴 E 𝐴)
2 epelg 5496 . . . 4 (𝐴𝐴 → (𝐴 E 𝐴𝐴𝐴))
32adantl 482 . . 3 (( E Fr 𝐴𝐴𝐴) → (𝐴 E 𝐴𝐴𝐴))
41, 3mtbid 324 . 2 (( E Fr 𝐴𝐴𝐴) → ¬ 𝐴𝐴)
54pm2.01da 796 1 ( E Fr 𝐴 → ¬ 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2106   class class class wbr 5074   E cep 5494   Fr wfr 5541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495  df-fr 5544
This theorem is referenced by:  tz7.2  5573  ordirr  6284
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